Proof of Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball

Each result cited is universally quantified over the data in its own statement. Write $\iota$ for the canonical map of The Canonical Map from the Natural Numbers to a Field, $\pi_{l}:\mathbb{R}^{q}\to\mathbb{R}$, $\pi_{l}(x)=x_{l}$, for the coordinate functions, and $W_{2}=\{y\in\mathbb{R}^{q}:\lVert y\rVert>2\}$. Continuity of a function on $\mathbb{R}^{q}$ is that of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (iv); by claim 1 of Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions it agrees with continuity at every point in the sense of Continuity at a Point for Maps Between Euclidean Spaces, which is the notion used in Functions of Class Cᵏ on an Open Subset of ℝⁿ. A function of class $C^{k+1}$ is of class $C^{1}$ and its partial derivatives are of class $C^{k}$, by clause 2 of Functions of Class Cᵏ on an Open Subset of ℝⁿ; a smooth function is of class $C^{k}$ for every $k$ (Smooth Functions on an Open Subset of ℝⁿ), so its partial derivatives are smooth as well. A function of class $C^{1}$ is continuous with continuous partial derivatives, by clause 1 of Functions of Class Cᵏ on an Open Subset of ℝⁿ.

For points $y,y'\in\mathbb{R}^{q}$ we use $y=y'+(y-y')$ and $\lVert y-y'\rVert=\lVert y'-y\rVert$ (claims 2 and 3 of Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, and claim 5 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square with $\lambda=-1$), together with $d_{E}(y,y')=\lVert y-y'\rVert$ (claim 2 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square).

Step 0: the derivatives of $\chi$ are bounded and vanish on $W_{2}$. The set $W_{2}$ is open: if $y\in W_{2}$ and $\lVert y'-y\rVert<\lVert y\rVert-2$, then $\lVert y\rVert\le\lVert y'\rVert+\lVert y-y'\rVert$ by claim 6 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square applied to $y=y'+(y-y')$, so $\lVert y'\rVert>2$; this says that the open ball of centre $y$ and radius $\lVert y\rVert-2$ lies in $W_{2}$, so $W_{2}$ is open in the metric sense, hence open by Euclidean Openness Agrees with Metric Openness on $\mathbb{R}^n$. Let $u:\mathbb{R}^{q}\to\mathbb{R}$ be of class $C^{1}$ with $u=0$ on $W_{2}$ and let $l\in[q]$. Then $u|_{W_{2}}=0\cdot u|_{W_{2}}$ is of class $C^{1}$ on $W_{2}$ by claim 3 of Restriction of a Cᵏ Function to an Open Subset, so by the scalar-multiple rule of claim 1 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions, $\partial_{l}(u|_{W_{2}})=0\cdot\partial_{l}(u|_{W_{2}})=0$ on $W_{2}$, and by claim 1 of Restriction of a Cᵏ Function to an Open Subset, $\partial_{l}u=0$ on $W_{2}$. Applying this to $u=\chi$ and then to $u=\partial_{i}\chi$ (which is smooth, hence of class $C^{1}$, and vanishes on $W_{2}$ by the first application), we get $\partial_{i}\chi=0$ and $\partial_{j}\partial_{i}\chi=0$ on $W_{2}$ for all $i,j\in[q]$. These functions are continuous on $\mathbb{R}^{q}$ (as $\chi$ is of class $C^{2}$) and vanish whenever $\lVert y\rVert>2$, so they are compactly supported by claim 2 of Compact Support on $\mathbb{R}^n$ Means Vanishing Outside a Bounded Set and bounded by claim 1 of A Continuous Compactly Supported Function on $\mathbb{R}^n$ is Bounded and Integrable. For $i,j\in[q]$ choose bounds $B_{i}$ and $B_{ji}$, nonnegative real numbers by that definition, with $|\partial_{i}\chi(y)|\le B_{i}$ and $|\partial_{j}\partial_{i}\chi(y)|\le B_{ji}$ for every $y$, and put

$$M_{1}=\sum_{i=1}^{q}B_{i},\qquad M_{2}=\sum_{i=1}^{q}\Bigl(\sum_{j=1}^{q}B_{ji}\Bigr),$$

finite sums of nonnegative real numbers. By claim 6 of Properties of Finite Sums (each summand of a sum of nonnegative terms is at most the sum, applied once for $M_{1}$ and twice for $M_{2}$), $B_{i}\le M_{1}$ and $B_{ji}\le M_{2}$ for all $i,j$, and $M_{1},M_{2}\ge0$ by claim 5 there.

Step 1: claim 1. Fix $R>0$; $R^{-1}>0$ by claim 7 of Elementary Order Arithmetic in an Ordered Field, so $|R^{-1}|=R^{-1}$ by Absolute Value in an Ordered Field, and claim 5 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square gives $\lVert R^{-1}x\rVert=R^{-1}\lVert x\rVert$ for every $x$. Multiplying by $R^{-1}$, respectively by $R$, using claim 5 of Elementary Arithmetic in an Ordered Field for the weak inequalities and claim 10 of Elementary Order Arithmetic in an Ordered Field for the strict one, together with $R\,R^{-1}=1$, we get for every $x$:

$$\lVert x\rVert\le R\iff\lVert R^{-1}x\rVert\le1,\qquad \lVert x\rVert\ge2R\iff\lVert R^{-1}x\rVert\ge2,\qquad \lVert x\rVert>2R\iff\lVert R^{-1}x\rVert>2 .$$

Consequently $0\le\chi_{R}\le1$, $\chi_{R}(x)=1$ whenever $\lVert x\rVert\le R$, $\chi_{R}(x)=0$ whenever $\lVert x\rVert\ge2R$, and $\chi_{R}$ is compactly supported by claim 2 of Compact Support on $\mathbb{R}^n$ Means Vanishing Outside a Bounded Set.

Apply Partial Derivatives, Continuity and $C^k$ Regularity under a Scaling Substitution with $n=q$, $m=1$, $c=0_{\mathbb{R}^{q}}$, $\lambda=R^{-1}$, $\mu=1$, $U=\mathbb{R}^{q}$ (open by claim 1 of Euclidean Space Is Open in Itself) and $f=\chi$: since $0_{\mathbb{R}^{q}}+R^{-1}x=R^{-1}x$ (claim 1 of Euclidean Space $\mathbb{R}^n$ is a Real Vector Space), the function $g$ there is $\chi_{R}$ and $V=\mathbb{R}^{q}$. By claim 5 there $\chi_{R}$ is smooth, and by claim 2 there, for every $x$ and $i\in[q]$,

$$\partial_{i}\chi_{R}(x)=R^{-1}\,\partial_{i}\chi(R^{-1}x).$$

Apply the same lemma with the same $c,\lambda,\mu,U$ to $f=\partial_{i}\chi$ (smooth): the function $g_{i}(x)=\partial_{i}\chi(R^{-1}x)$ has, by claim 2 there, $\partial_{j}g_{i}(x)=R^{-1}\partial_{j}\partial_{i}\chi(R^{-1}x)$ for every $x$ and $j\in[q]$. Since $\partial_{i}\chi_{R}=R^{-1}g_{i}$ on $\mathbb{R}^{q}$, the scalar-multiple rule of claim 1 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions gives

$$\partial_{j}\partial_{i}\chi_{R}(x)=R^{-1}\,\partial_{j}g_{i}(x)=R^{-2}\,\partial_{j}\partial_{i}\chi(R^{-1}x),$$

where $R^{-2}=R^{-1}R^{-1}$. By Step 0 and claim 4 of Properties of the Absolute Value in an Ordered Field, $|\partial_{i}\chi_{R}(x)|\le R^{-1}B_{i}\le M_{1}R^{-1}$ and $|\partial_{j}\partial_{i}\chi_{R}(x)|\le R^{-2}B_{ji}\le M_{2}R^{-2}$ (claim 5 of Elementary Arithmetic in an Ordered Field). This proves claim 1. We record for Step 3 that if $\lVert x\rVert>2R$ then $R^{-1}x\in W_{2}$, so by Step 0

$$\chi_{R}(x)=0,\qquad \partial_{i}\chi_{R}(x)=0,\qquad \partial_{j}\partial_{i}\chi_{R}(x)=0\qquad(\lVert x\rVert>2R).$$

Step 2: the function $s(x)=\tfrac{1}{2}\lVert x\rVert^{2}$. By claim 2 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square, $s(x)=\tfrac{1}{2}\,d_{E}(x,0_{\mathbb{R}^{q}})^{2}$, so A Scaled Squared Distance to a Point is of Class $C^2$, with Gradient and Hessian applies with $n=q$, $U=\mathbb{R}^{q}$, $a=0_{\mathbb{R}^{q}}$, $c=\tfrac{1}{2}$ and its function $q$ taken to be $s$: by its claim 1, $\partial_{j}s(x)=2\cdot\tfrac{1}{2}\,(x_{j}-0)=x_{j}$, that is $\partial_{j}s=\pi_{j}$; by its claim 2, $s$ is of class $C^{2}$; and by its claim 3, $D^{2}s(x)=(2\cdot\tfrac{1}{2})I_{q}=I_{q}$ (the matrix scalar multiple being that of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (i)), so by The Hessian Matrix of a Function of Class C², $\partial_{j}\partial_{i}s(x)=(I_{q})_{ji}=\delta_{ji}$ for all $i,j\in[q]$ and all $x$. Moreover $s$ is smooth: $x\mapsto\lVert x\rVert^{2}$ is smooth by The Squared Euclidean Norm is Smooth, and $s$ is its scalar multiple by $\tfrac{1}{2}$, smooth by claim 3 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions.

Step 3: claim 2. Fix $R>0$. As $\psi_{R}=\chi_{R}\,s$ is a product of smooth functions, it is smooth by claim 3 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions; it vanishes wherever $\chi_{R}$ does, hence whenever $\lVert x\rVert\ge2R$, so it is compactly supported by claim 2 of Compact Support on $\mathbb{R}^n$ Means Vanishing Outside a Bounded Set. By the product rule of claim 1 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions and Step 2, for all $x$ and $i,j\in[q]$,

$$\partial_{i}\psi_{R}(x)=\partial_{i}\chi_{R}(x)\,s(x)+\chi_{R}(x)\,x_{i},$$

and, applying the sum and product rules of the same claim to this expression (whose two summands are products of functions of class $C^{1}$), with $\partial_{j}\pi_{i}=\partial_{j}\partial_{i}s=\delta_{ji}$ by Step 2,

$$\partial_{j}\partial_{i}\psi_{R}(x)=\partial_{j}\partial_{i}\chi_{R}(x)\,s(x)+\partial_{i}\chi_{R}(x)\,x_{j}+\partial_{j}\chi_{R}(x)\,x_{i}+\chi_{R}(x)\,\delta_{ji}.$$

Put $M=2M_{2}+4M_{1}+1$, which does not depend on $R$; since $M_{1},M_{2}\ge0$ (Step 0) and $0\le1$ (claim 1 of Elementary Arithmetic in an Ordered Field), claim 2 there gives $0\le M$ and $M_{1}+1\le M$, and $0\le\lVert x\rVert$ for every $x$ by claim 1 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square. Fix $x$ and $i,j$. If $\lVert x\rVert>2R$, all terms on the right of both displays vanish by Step 1, so $\partial_{i}\psi_{R}(x)=0$ and $\partial_{j}\partial_{i}\psi_{R}(x)=0$, and the two bounds hold because their right sides are nonnegative ($0\le M\lVert x\rVert$ by claim 5 of Elementary Arithmetic in an Ordered Field). If $\lVert x\rVert\le2R$, then $s(x)=\tfrac{1}{2}\lVert x\rVert\lVert x\rVert\le\tfrac{1}{2}\,2R\lVert x\rVert=R\lVert x\rVert$ and $s(x)\le\tfrac{1}{2}(2R)^{2}=2R^{2}$ (claim 5 of Elementary Arithmetic in an Ordered Field and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field), $|x_{i}|,|x_{j}|\le\lVert x\rVert\le2R$ by claim 4 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square, and $|\chi_{R}(x)|\le1$. Using claim 1, the multiplicativity and the triangle inequality of the absolute value (claims 4 and 5 of Properties of the Absolute Value in an Ordered Field) and claim 5 of Elementary Arithmetic in an Ordered Field,

$$|\partial_{i}\psi_{R}(x)|\le M_{1}R^{-1}\cdot R\lVert x\rVert+\lVert x\rVert=(M_{1}+1)\lVert x\rVert\le M\lVert x\rVert,$$ $$|\partial_{j}\partial_{i}\psi_{R}(x)|\le M_{2}R^{-2}\cdot2R^{2}+M_{1}R^{-1}\cdot2R+M_{1}R^{-1}\cdot2R+1=2M_{2}+4M_{1}+1=M .$$

Finally, $\Delta\psi_{R}(x)=\sum_{i=1}^{q}\partial_{i}\partial_{i}\psi_{R}(x)$ satisfies $|\Delta\psi_{R}(x)|\le\sum_{i=1}^{q}|\partial_{i}\partial_{i}\psi_{R}(x)|$ by claim 2 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers, and $\sum_{i=1}^{q}|\partial_{i}\partial_{i}\psi_{R}(x)|\le\sum_{i=1}^{q}M$ by claim 1 of the same lemma; and $\sum_{i=1}^{q}M=M\,\iota(q)=qM$ by claim 3 of Properties of Finite Sums and The Canonical Map from the Natural Numbers to a Field. This proves claim 2.

Step 4: claim 3. Fix $R>0$ and let $B=\{y\in\mathbb{R}^{q}:\lVert y\rVert<R\}$. The set $B$ is open: if $y\in B$ and $\lVert y'-y\rVert<R-\lVert y\rVert$, then $\lVert y'\rVert\le\lVert y\rVert+\lVert y'-y\rVert<R$ by claim 6 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square applied to $y'=y+(y'-y)$, so the argument of Step 0 applies. By Step 1, $\chi_{R}(y)=1$ for every $y\in B$, so $\psi_{R}(y)=s(y)$ for every $y\in B$; that is, $\psi_{R}|_{B}=s|_{B}$. Let $x\in B$ and $i,j\in[q]$. By claim 1 of Restriction of a Cᵏ Function to an Open Subset applied to $\psi_{R}$ and to $s$ (both of class $C^{1}$ on $\mathbb{R}^{q}$),

$$\partial_{i}\psi_{R}(x)=\partial_{i}(\psi_{R}|_{B})(x)=\partial_{i}(s|_{B})(x)=\partial_{i}s(x)=x_{i},$$

the last equality by Step 2. Since this holds at every point of $B$, the functions $\partial_{i}\psi_{R}$ and $\partial_{i}s$ (both of class $C^{1}$ on $\mathbb{R}^{q}$, as $\psi_{R}$ and $s$ are of class $C^{2}$) have the same restriction to $B$, and the same claim 1 gives $\partial_{j}\partial_{i}\psi_{R}(x)=\partial_{j}\partial_{i}s(x)=\delta_{ji}=\delta_{ij}$. Hence $\Delta\psi_{R}(x)=\sum_{i=1}^{q}\delta_{ii}=\sum_{i=1}^{q}1=\iota(q)=q$ by The Canonical Map from the Natural Numbers to a Field, and $\psi_{R}(x)=s(x)=\tfrac{1}{2}\lVert x\rVert^{2}$. This proves claim 3.